# Experimental and Numerical Thickness Analysis of TRIP Steel under Various Degrees of Deformation in Bulge Test

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## Abstract

**:**

## 1. Introduction

^{2}emissions. It is for the above-mentioned reasons that steel sheets with higher strength properties, which are characterized by good plasticity, have been widely used in recent years in the production of car bodies [1,2,3,4,5,6].

## 2. Materials and Methods

#### 2.1. Material

#### 2.2. Hydraulic Bulge Test

_{0}= 80 mm). The test specimens were deformed gradually to a height corresponding to the deformation for determining the strain-hardening exponent by a uniaxial tensile test (5–15%). The samples were labeled sequentially from T8 to T3. Samples with T8 designation were deformed up to 5.2 mm bulge height, samples T7 were deformed up to 9.9 mm, samples T6 were deformed up to 13.5 mm, samples T5 were deformed up to 14.4 mm, samples T4 were deformed up to 18.8 mm and samples T3 were deformed up to 19.3 mm.

#### 2.3. Measurement of Sample Thickness Using a Digital Point Micrometer

#### 2.4. Measurement of Sample Thickness Using the Optical Measuring System ARGUS

#### 2.5. Numerical Simulation of the Bulge Test in FEM Software

#### 2.5.1. Yield Surface

#### 2.5.2. Hardening Law

- Hollomon$$\sigma =K\mathrm{\xb7}{\phi}^{n}$$
- Krupkowski$$\sigma =K\mathrm{\xb7}{\left({\phi}_{0}+{\phi}_{pl}\right)}^{n}$$
_{0}is the pre-strain and φ_{pl}represents the plastic strain. Material model constants used in both hardening rules can be seen in Table 4.

## 3. Results

#### 3.1. Experimental Bulge Test Results

#### 3.2. Simulation Bulge Test Results

_{sim}is the simulation thickness value and X

_{exp}is the experimental thickness value.

## 4. Discussion

## 5. Conclusions

- Hollomon’s hardening law exhibited better accuracy in predicting thickness in comparison with the Krupkowski hardening law in most cases.
- The Hill90 yield criterion was found to be more suitable for simulation of bulge test using TRIP steel.
- Software with implicit solver and triangular shell elements showed better accuracy in predicting thickness of deformed sheet in bulge test than software with explicit solver and hexagonal shell elements in most cases.
- The degree of deformation impacted the simulation accuracy, while thickness prediction of samples with greater deformation was less accurate in comparison with samples with lower deformation and lower bulge height.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The uniaxial tensile test: Testing machine TIRAtest 2300 (

**a**); Specimen dimensions for the test (mm) (

**b**).

**Figure 13.**Comparison of predicted thinning values of samples T3 (

**left**) and T6 (

**right**) using software with explicit solver in program environment.

**Figure 14.**Comparison of predicted and measured thickness values of sample T3 using different types of yield criteria, hardening laws and integration scheme.

**Figure 15.**Comparison of predicted and measured thickness values of sample T4 using different types of yield criteria, hardening laws and integration scheme.

**Figure 16.**Comparison of predicted and measured thickness values of sample T5 using different types of yield criteria, hardening laws and integration scheme.

**Figure 17.**Comparison of predicted and measured thickness values of sample T6 using different types of yield criteria, hardening laws and integration scheme.

**Figure 18.**Comparison of predicted and measured thickness values of sample T7 using different types of yield criteria, hardening laws and integration scheme.

**Figure 19.**Comparison of predicted and measured thickness values of sample T8 using various yield criteria, hardening laws, and integration scheme.

C (%) | Mn (%) | Si (%) | P (%) | S (%) | Al [%] | Nb (%) | Ti (%) | Mo (%) | Cr (%) |
---|---|---|---|---|---|---|---|---|---|

0.204 | 1.683 | 0.200 | 0.018 | 0.003 | 1.730 | 0.004 | 0.009 | 0.008 | 0.055 |

Dir. (°) | E (GPa) | R_{p0.2}(MPa) | R_{m}(MPa) | A_{80}(%) | r (-) | r_{m}(-) | Δr (-) | n (-) | n_{m}(-) | Δn (-) |
---|---|---|---|---|---|---|---|---|---|---|

0 | 441 | 766 | 27.9 | 0.680 | 0.293 | |||||

45 | 210 | 442 | 762 | 25.4 | 0.805 | 0.804 | −0.003 | 0.294 | 0.290 | −0.009 |

90 | 450 | 766 | 25.9 | 0.926 | 0.278 |

_{p0.2}= yield stress, R

_{m}= ultimate tensile strength, A

_{80}= total elongation, r = plastic strain ratio, n = strain-hardening exponent, n

_{m}= average value of strain-hardening exponent, r

_{m}= average value of plastic strain ratio Δr = planar anisotropy of plastic strain ratio, and Δn = planar anisotropy of strain-hardening exponent.

r_{0}(-) | r_{45}(-) | r_{90}(-) | σ_{0}(MPa) | σ_{45}(MPa) | σ_{90}(MPa) | σ_{biax}(-) |
---|---|---|---|---|---|---|

0.680 | 0.805 | 0.926 | 441 | 442 | 450 | 1.01 |

Model | K (MPa) | n (-) | φ_{0}(-) |
---|---|---|---|

Hollomon | 1331 | 0.219 | - |

Krupkowski | 1336 | 0.227 | 0.00765 |

**Table 5.**The mean deviations (%) of predicted thickness values from the experimental thickness values using software with the explicit solver.

Sample | Hill48 Hollomon | Hill48 Krupkowski | Hill90 Hollomon | Hill90 Krupkowski |
---|---|---|---|---|

T3 | 11.39 | 12.97 | 2.20 | 3.72 |

T4 | 11.00 | 11.94 | 2.84 | 3.85 |

T5 | 5.41 | 5.98 | 1.06 | 1.57 |

T6 | 6.81 | 6.90 | 2.83 | 3.32 |

T7 | 2.82 | 3.12 | 0.65 | 1.18 |

T8 | 0.24 | 0.38 | 0.19 | 0.28 |

**Table 6.**The mean deviations (%) of predicted thickness values from the experimental thickness values using software with the implicit solver.

Sample | Hill48 Hollomon | Hill48 Krupkowski | Hill90 Hollomon | Hill90 Krupkowski |
---|---|---|---|---|

T3 | 2.08 | 2.59 | 2.41 | 1.97 |

T4 | 2.40 | 2.60 | 1.90 | 2.18 |

T5 | 2.05 | 2.43 | 1.97 | 1.94 |

T6 | 0.88 | 0.90 | 0.84 | 0.79 |

T7 | 0.55 | 0.78 | 0.45 | 0.53 |

T8 | 0.42 | 0.42 | 0.37 | 0.29 |

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**MDPI and ACS Style**

Spišák, E.; Majerníková, J.; Kaščák, Ľ.; Mulidrán, P.; Rohaľ, V.; Bidulský, R.
Experimental and Numerical Thickness Analysis of TRIP Steel under Various Degrees of Deformation in Bulge Test. *Materials* **2022**, *15*, 2299.
https://doi.org/10.3390/ma15062299

**AMA Style**

Spišák E, Majerníková J, Kaščák Ľ, Mulidrán P, Rohaľ V, Bidulský R.
Experimental and Numerical Thickness Analysis of TRIP Steel under Various Degrees of Deformation in Bulge Test. *Materials*. 2022; 15(6):2299.
https://doi.org/10.3390/ma15062299

**Chicago/Turabian Style**

Spišák, Emil, Janka Majerníková, Ľuboš Kaščák, Peter Mulidrán, Vladimír Rohaľ, and Róbert Bidulský.
2022. "Experimental and Numerical Thickness Analysis of TRIP Steel under Various Degrees of Deformation in Bulge Test" *Materials* 15, no. 6: 2299.
https://doi.org/10.3390/ma15062299